Problem 150 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ (-\infty,-1] \cap[-4, \infty)=[-4,-1] $$

Step-by-Step Solution

Answer

The statement that \((-\infty,-1] \cap [-4, \infty)=[-4,-1]\) is true.

Step 1: Understanding the notation

To tackle this exercise, there's a need to understand what the symbols mean. The interval \((-\infty,-1]\) signifies all real numbers from negative infinity up to and including -1. Similarly, the interval \([-4, \infty)\) means all real numbers from -4 up to positive infinity. The \(\cap\) symbol represents the intersection of these two sets, meaning the common elements between them.

Step 2: Determining the intersection

The intersection of this two intervals can be visualized on a number line. Since \([-4, \infty)\) includes all numbers from -4 to positive infinity and \((-\infty,-1]\) includes all numbers from negative infinity up to and including -1, their intersection, or common elements, will be the interval from -4 to -1, inclusive. Therefore, \((-\infty,-1] \cap [-4, \infty)\) equals \([-4, -1]\).

Step 3: Evaluating the original statement

Having determined the correct intersection of the given intervals in step 2, this can be compared to the intersection provided in the exercise. The exercise states that the intersection of the sets \((-\infty,-1]\) and \([-4, \infty)\) is \([-4,-1]\), which is the same one that was obtained when determining the intersection manually.

Key Concepts

Set Theory in Mathematics

Set theory is a fundamental part of mathematics that deals with the collection of objects called 'sets.' These objects can be anything: numbers, letters, symbols, or even other sets. The beauty of set theory lies in its ability to provide a foundation for virtually every other part of mathematics by studying the properties and relations of sets.

In simple terms, set theory allows mathematicians to define and work with different groups of elements. A crucial concept within set theory is the idea of 'intersection,' represented by the symbol \(\cap\). The intersection of two sets is a new set containing all the elements that are common to both sets provided. For example, if Set A includes {1, 2, 3} and Set B has {2, 3, 4}, their intersection Set A \(\cap\) Set B would be {2, 3}.

Dealing with the infinite nature of sets, as often occurs with the real numbers, set theory uses different kinds of notations to represent ranges, including interval notation, which is key to understanding the solutions to many mathematical problems like the one in our exercise.

Real Numbers Interval

An interval in the context of real numbers is a way of denoting a range of numbers between two endpoints. These endpoints could be actual real numbers or the concept of infinity. Interval notation is a concise form of writing this range and is incredibly important in various fields of calculus, algebra, and beyond.

Types of Intervals

There are several types of intervals based on whether the endpoints are included in the set:

• Closed interval [a, b]: Both a and b are included in the set.
• Open interval (a, b): Neither a nor b are included.
• Half-open (or half-closed) interval: In (a, b] or [a, b), only one of the endpoints is included.

In our textbook exercise, we dealt with a half-open interval \( (-\infty,-1] \) and an open interval on one end \( [-4, \infty) \). The notation used here tells a story - it provides a visual shorthand that mathematicians and students can read to understand the bounds and inclusivity of a range of numbers at a glance.

Number Line Visualization

Visualization using a number line is an indispensable tool for understanding the concepts of set theory, especially when dealing with intervals. On a number line, every point corresponds to a real number, which allows us to visually represent sets of numbers, be they finite or infinite.

When considering the intersection of two intervals on a number line, we essentially look for the overlap region where both intervals 'exist.' As demonstrated in the solution of the exercise, plotting the intervals \( (-\infty,-1] \) and \( [-4, \infty) \) helps to discover that they intersect from -4 to -1. This method not only confirms the exact elements within the intersection but also helps eliminate any confusion about whether the endpoints are included or excluded, which is a frequent source of error in interval notation problems.

Hence, employing a number line gives us a remarkable way to 'see' mathematics and allows a clearer understanding of the relationship between different sets; in this case, it perfectly illustrated the intersection between the two given intervals.